Gate-tunable quantum pathways of high harmonic generation in graphene

Under strong laser fields, electrons in solids radiate high-harmonic fields by travelling through quantum pathways in Bloch bands in the sub-laser-cycle timescales. Understanding these pathways in the momentum space through the high-harmonic radiation can enable an all-optical ultrafast probe to observe coherent lightwave-driven processes and measure electronic structures as recently demonstrated for semiconductors. However, such demonstration has been largely limited for semimetals because the absence of the bandgap hinders an experimental characterization of the exact pathways. In this study, by combining electrostatic control of chemical potentials with HHG measurement, we resolve quantum pathways of massless Dirac fermions in graphene under strong laser fields. Electrical modulation of HHG reveals quantum interference between the multi-photon interband excitation channels. As the light-matter interaction deviates beyond the perturbative regime, elliptically polarized laser fields efficiently drive massless Dirac fermions via an intricate coupling between the interband and intraband transitions, which is corroborated by our theoretical calculations. Our findings pave the way for strong-laser-field tomography of Dirac electrons in various quantum semimetals and their ultrafast electronics with a gate control.


1.
The authors claimed that they observed the lightwave-driven dynamics of massless Dirac fermions, but the laser intensity of 3.1 GW/cm2 was adopted in measurement shown in Fig. 2-3. Unfortunately it is not in the non-perturbative regime. As displayed in Fig. 1e, the intensity of the 5th harmonic scales as 5 ∝ 5 @ = 3.1 GW/cm2, in which multiphoton excitation dominates the nonlinear current and gives perturbative harmonic generation. The authors may check if the experimental observation can be reproduced in the framework of nonlinear optics by taking into account the fifth-order susceptibility.
In the intense laser field regime, strong field ionization, such as Zenner tunneling [Science 356, 736 (2017); Nature 550, 224 (2017)], the subsequent acceleration of electron-hole pair and generalized recollision lead to the emission of high harmonics [Phys. Rev. Lett. 113, 073901 (2014)]. In this work the laser field strength may not be the primary driving force of Dirac electron dynamics, but the multi-photon transitions.
The enhancement of the 5th harmonic, H5, is attribute to the partial closing of competing transition channels due to Pauli blocking. To this extent, the closing of interference channels dose not possess lightwave-driven character. Therefore, whether the present finding can be called as lightwave-driven Dirac fermions is open to debate, and the authors should give more convincing arguments on this.

2.
In Fig. 2c and d, there exists a dip around 2||=0.4eV on the curve of 5 versus chemical potential. Why is that? Should this phenomenon be attribute to selectively closing of some interference channels, or the jitter of the measurements? Also, note that the dip is absent in Fig.  S4b, please clarify.

3.
The number of photons involved in multiphoton processes should be an integer. It is interesting to notice that the resonance-like profiles of 5 peaks at 2|| ph = 3.4. Intuitively, the Pauli blocking should take place once 2|| exceeds 3ph. Could the authors give an explanation?

4.
Is there a connection between the linear energy dispersion of the Dirac cone and the observed modulation of HHG? If the Dirac band replaced by other types of dispersion, such as parabolic band with a narrower bandgap, can the unique ellipticity dependence of HHG still be observed?

5.
Some key experimental parameters are not given in the manuscript, such as, dimensions of the graphene sample between the electrodes, and the focal spot size of the pump pulses. Will the H5 be affected if the focused laser illuminates on the graphene-metal junction? How to exclude that influence? 6. Figure 1e shows that the pump intensity of =3.1 GW/cm2 can be viewed as a starting point where electron dynamics changes from perturbative to non-perturbative. This transition region has not been well explored previously, and the competition between the multiphoton excitation and lightwave-driven electron dynamics ought to be revealed. I would be happy to see it being clarified.
Reviewer #2: Remarks to the Author: In their work, Cha et al. present an interesting experimental work on high-harmonic generation (HHG) in graphene monolayer, supported by theoretical simulations. They investigate the role of chemical doping on the HHG in graphene and demonstrate interesting experimental findings, in particular the role of the chemical doping on the polarization state of the emitted harmonics. Because of the specific nature of the Dirac cone, the authors show that the interband excitation channel can be suppressed in the strong doping case, thus affecting the harmonic emission. This work is timely and interesting and deserves publication. However, I find that the authors must address some points before the manuscript can be recommended for publication. These points are listed below: i) I find the title over-claiming the results of the manuscript. What the authors show is at best a dependence on the pathways. I felt to see in which aspect they are visualizing them. Especially, the understanding of the results can only be made because of the theoretical support, and the experiment alone cannot "visualize" anything.
ii) I think that the explanation of the forbidden excitation channel is related to some prior works. In particular, in condensed matter, there is a quantity called the joint-density-of-state, which is exactly related to the quantity of available optical transitions. Maybe the authors could refer to this quantity and compute it for the different doping, as done for instance in [Phys. Rev. Lett. 118, 087403 (2017)] or in [Nature Photonics volume 14, pages 183-187 (2020)].
iii) While the results are interesting, I find the analysis of the results a bit weak, especially for Fig.  2. Following the argument in introduction, one would expect an abrupt change of the yield versus doping at exactly 2µ/E_ph=1. However, around this value, almost nothing is happening. I think that the authors should explain this better.
iv) The same figure displays a maximum around 3.4, as nicely shown by the authors for various samples. However, again, this value is not explained. At the moment, we do not know if this value is due to a material's property, a laser property, or a combination of both. I think that the simulations here could come at help. The authors could also simulate the results of Fig. 2 and try to explain the physical origin of the maximum at 3.4. Which mechanism is dominant for instance. In fact, the same plot as Fig. 2c-d, but decomposed into interband and intraband could really provide some insights here. I would suggest the authors to add this simulations.

General remarks of Reviewer 1:
The manuscript entitled "Imaging quantum pathways of lightwave-driven massless Dirac fermions" by Soonyoung Cha et al. presents an interesting work on high harmonic generation (HHG) from graphene embedded in a field-effect transistor device. The authors measured HHG intensity and polarization as a function of hole doping level, driven by linearly and elliptically polarized mid-infrared laser pulses. They showed that multiphoton excitation channels are destructively interfered and can be partially closed by tuning the chemical potential. In addition, they found that the complex dynamics between the interband and intraband transitions can be decoupled, according to the dependence of HHG polarization states on the laser ellipticity. The authors suggested that quantum pathways of massless Dirac fermions in graphene can be mapped out through HHG spectroscopy. The manuscript reports a beautiful experiment, but I have several questions and concerns on the viewpoints.

Our reply:
We deeply appreciate the referee for her/his effort to review our manuscript. In addition, we thank for her/his appreciation of the importance of our work and valuable feedbacks which have been tremendously helpful to improve our manuscript. We have addressed her/his questions and concerns carefully as below.

Original comment (2):
1. The authors claimed that they observed the lightwave-driven dynamics of massless Dirac fermions, but the laser intensity of 3.1 GW/cm 2 was adopted in measurement shown in Fig. 2-3. Unfortunately it is not in the non-perturbative regime. As displayed in Fig. 1e

Our reply:
We appreciate the reviewer for her/his critical comment on the interpretation of our experimental results. We totally agree with the reviewer that enhancement of through multi-photon transitions can be well-explained by light-matter interaction in the perturbative regime. However, although it is not completely in the non-perturbative regime, our results at 3.1 GW cm -2 reports emerging new features in HHG which cannot be explained by the perturbative electron dynamics.
First of all, our apology for the unclear presentation of Fig. 1e in the previous version. For the laser intensity of 3.1 GW cm -2 and its neighboring intensity range, the 5 th harmonic intensity roughly scales as ∝ ~ . (Fig. R1), which indicates the light-matter interaction is deviating from the perturbative regime. We have added a black arrow to indicate the specific data point clearly in the log-log plot in the revised manuscript. In addition, under elliptically polarized excitation, the polarization axis of the 5 th harmonics rotates as the laser intensity increases from 2.4 GW cm -2 to 10.1 GW cm -2 ( Fig. 1g) while the ellipticity of the laser pulse is fixed at 0.3. This rotation of the polarization axis is clear indication of the non-perturbative response at 3.1 GW cm -2 . In nonlinear optics in the perturbative regime, where the fifth-order susceptibility are independent on the laser intensity, the polarization axis cannot rotate as the intensity increases. We assume that the perturbative contribution from higher-order susceptibility on the fifth harmonic generation is negligible considering more than an order magnitude weaker 7 th harmonic generation as shown in Fig. 1d which is taken even at 50 GW cm -2 .
In the perturbative nonlinear optics, the intensity of a fifth harmonic signal 5ω under laser-field ω can generally be calculated by the following equation Thus, this indicates that the polarization axis of high harmonics cannot rotate over intensity in the perturbative regime.
More interestingly, in Fig. 3b and 3c, rotation direction of the polarization axis (either clockwise or counter-clockwise) can be controlled by the helicity of the elliptically polarized laser (i.e. the relative phase of the laser fields along the x-and y-directions).
Therefore, we believe that the non-perturbative electron dynamics significantly contribute to our experimental data ( Fig. 2 and 3) measured at 3.1 GW cm -2 although the perturbative electron dynamics such as the multi-photon transitions contributes strongly as well.
However, we totally agree with the reviewer that more rigorous studies including systematic control of carrier envelop phase (CEP) of the laser field are required to claim our observed phenomena as the lightwave-driven phenomena. We have removed our expression "lightwave-driven" referring to our observed phenomena in graphene and included the discussion above in the revised main text and supplementary information following the reviewer's comment.

Original comment (3):
2. In Fig. 2c and d, there exists a dip around 2| | = 0.4 eV on the curve of versus chemical potential. Why is that? Should this phenomenon be attribute to selectively closing of some interference channels, or the jitter of the measurements? Also, note that the dip is absent in Fig. S4b, please clarify.

Our reply:
We thank the reviewer for her/his effort to carefully check our experimental data and provide us valuable comments which encourage us to obtain deeper understanding on the chemical potential dependence of the 5 th harmonic signal. In fact, experimental data in Fig  Interestingly, our theoretical calculation (Fig. R3) also shows a dip-like profile around at similar chemical potential. As dephasing time T2 decreases from 4 fs to 2 fs, the dip-like profile becomes more pronounced while the resonance-like profile (originating from channels via multiphoton transitions) is heavily suppressed. In fact, the theoretical calculation reveals that two characteristic profiles are combined in as a function of chemical potential: the resonance-like profile at 2|μ|/ = 3.4 and the plateau-like profile at 2|μ|/ < 2. While the resonance-like profile is originated from multi-photon transition channels, the physical origin of the plateau-like profile is not clear at the moment.
Identifying the physical origin of this profile requires comprehensive experimental and theoretical investigation with systematic control of laser intensity and sample qualities which will allow to quantify and to control dephasing time. Although this will be certainly of our future interest, unfortunately, this goes beyond the scope of the current study. We have included the discussion above in the main text and the supplementary information of the revised manuscript.

Figure. R3. Calculated -dependent HHG with different dephasing time (T 2 ).
Harmonic intensity is given by for the nth order harmonic intensity. Harmonic intensity is calculated with T 2 of 4 fs (blue curve), 3 fs (red curve) and 2 fs (black curve).

Original comment (4):
3. The number of photons involved in multiphoton processes should be an integer. It is interesting to notice that the resonance-like profiles of peaks at | | = 3.4. Intuitively, the Pauli blocking should take place once 2| | exceeds 3 ph. Could the authors give an explanation?
Our reply: We appreciate the reviewer for raising this important question on the resonance-like profile of which shows the maximum intensity at 2|μ|/ = non-integer. As the reviewer points out, the number of photons involved in multiphoton transition processes is an integer which will show series of multiple sharp resonance-like profiles at 2|μ|/ = n (n is an integer). In the perturbative regime without considering the scattering process, such profile has been theoretically predicted for by the recent theoretical study [PRB 99, 195407(2019)], which we have drawn in Fig. R4 using their results. Depending on the photon number required for multi-photon transitions and the azimuthal angle, Dirac states generate with characteristic sign and magnitude, which destructively interfere when radiating . As 2|μ| increases, Pauli blocking sequentially disables contribution from the Dirac states starting from near the Dirac point to higher energy. As shown in Fig. R5a, increases as the destructive interference is partly eliminated while eventually disappears as all the resonant Dirac states are disabled.
Our theoretical calculation (Fig. R5a) also shows that the maximum intensity is exhibited around at 2|μ|/ = 3.4, which is the most common value for 2|μ|/ in our experimental data (Fig.S5). We have included the discussion above in the revised manuscript.

Our reply:
We appreciate the reviewer for her/his critical question on a connection between the linear energy dispersion and the ellipticity dependence of HHG in graphene. In the regime where carriers are dominantly excited by multi-photon transitions, our studies show that the ellipticity dependence of HHG originates from the nonlinear coupling between interband and intraband transitions. For the linear energy band dispersion from = ± • , linearly polarized laser-field along the x-direction ( ) excites charge carriers in the peculiar anisotropic distribution in the momentum space as shown in Fig. 4a and 4b, which reflects the transition dipole moment texture of massless Dirac fermion (Fig. 4c). Under elliptically polarized laser-field, additional laser-field along the y-direction with a phase delay of /2 ( ) can modify the anisotropic distribution by driving charge carriers within the same bands. Our theoretical calculation (Fig. 4f) shows that this pathway ( + ) can generate efficiently.
In order to answer the reviewer's question, we have investigated the ellipticity dependence of HHG from the "massive" Dirac fermions by introducing a gap of ~ 600 meV which corresponds to ~ 2.2 . is the photon energy of laser excitation in our experiment.
Both the linear band dispersion (black solid line in Fig. R6c) and the anisotropic transition dipole moment texture (Fig. R6d) are relaxed in the cases of "massive" Dirac fermions (blue solid line in Fig. R6c and Fig. R6e). Under the same laser intensity and ellipticity, massive Dirac fermions generates significantly smaller (red solid line in Fig. R6b) in comparison to massless Dirac fermions (red solid line in Fig. R6a). This indicates that the observed ellipticity dependence of HHG is strongly connected to the linear band dispersion.
However, we note that considerable amount of can be still generated even from massive Dirac fermions. This is potentially because the anisotropy of the transition dipole moment texture ( Fig. R6e) is not completely relaxed.  shows that even parabolic bands can efficiently generate under elliptically polarized excitation.
Therefore, in the revised manuscript, we have changed our expression "unique mechanism for massless Dirac fermions" to "characteristic mechanism for massless Dirac fermions" in order to address the reviewer's concern.
Original comment (6): 5. Some key experimental parameters are not given in the manuscript, such as, dimensions of the graphene sample between the electrodes, and the focal spot size of the pump pulses.
Will the H5 be affected if the focused laser illuminates on the graphene-metal junction?
How to exclude that influence?

Our reply:
We appreciate the reviewer for her/his effort to carefully check our experimental configuration. Following the reviewer's comments, we have included information about dimension of the graphene samples and the focal spot size of pump pulses in the revised manuscript. The mid-infrared pulse is focused on graphene by a ZnSe objective lens, and the spot size of the pulse is measured as 150 m (full width at half-maximum of intensity, FWHM) by using knife-edge method. Our graphene device is shown in the Fig. R7 below.
The size of graphene device is typically a few millimeters by a few millimeters, which is wide enough to avoid the graphene-metal junction being illuminated by focused laser.
During the experiment, the mid-infrared laser beam is illuminated roughly at the center of graphene devices by monitoring position of a guiding HeNe laser (632 nm) beam spot which propagates along the same optical path of mid-infrared laser.

Figure. R7. Image of the graphene device on the slide glass. Graphene is located
between the source and drain contact, surrounded by red lines.
Original comment (7): 6. Figure 1e shows that the pump intensity of exc = 3.1 GW/cm2 can be viewed as a starting point where electron dynamics changes from perturbative to non-perturbative. This transition region has not been well explored previously, and the competition between the multiphoton excitation and lightwave-driven electron dynamics ought to be revealed. I would be happy to see it being clarified.

Our reply:
We appreciate the reviewer for providing us an invaluable comment that our pump intensity can be considered as a transitioning point where electron dynamics change from perturbative to non-perturbative. In the perturbative regime, generation of 5  investigation is experimentally challenging at this moment due to the long-term laser damage issue as shown in Fig. S2. HHG study with fewer cycle laser pulses can help to address the laser damage issue, which is of our future interest. We have included the discussion above in the introduction and conclusion parts of the revised manuscript.

Reply to Reviewer #2
Original comment (1): In their work, Cha et al. present an interesting experimental work on high-harmonic generation (HHG) in graphene monolayer, supported by theoretical simulations. They investigate the role of chemical doping on the HHG in graphene and demonstrate interesting experimental findings, in particular the role of the chemical doping on the polarization state of the emitted harmonics. Because of the specific nature of the Dirac cone, the authors show that the interband excitation channel can be suppressed in the strong doping case, thus affecting the harmonic emission. This work is timely and interesting and deserves publication. However, I find that the authors must address some points before the manuscript can be recommended for publication. These points are listed below: Our reply: We deeply appreciate the referee for her/his effort to review our manuscript.
In addition, we thank for her/his appreciation of the importance of our work and valuable comments on the experimental and theoretical results which have been tremendously helpful to improve our manuscript. We have addressed her/his questions and concerns carefully as below.
Original comment (2): 1. I find the title over-claiming the results of the manuscript. What the authors show is at best a dependence on the pathways. I felt to see in which aspect they are visualizing them.
Especially, the understanding of the results can only be made because of the theoretical support, and the experiment alone cannot "visualize" anything.

Our reply:
We appreciate the reviewer for her/his critical comment on the title of our manuscript. We agree with the reviewer that our expressions such as "imaging" or "visualizing" would not be appropriate. Although our experimental results combined with the theoretical support provide critical information to resolve pathways of high harmonic generation in graphene, we do not physically image their trajectory either in the momentum space or the real space.
Following the reviewer's comment, we have changed the title of our manuscript to 'Gatetunable quantum pathways of high harmonic generation in graphene' and revised all the related descriptions in the main text.

Original comment (3):
2. I think that the explanation of the forbidden excitation channel is related to some prior works. In particular, in condensed matter, there is a quantity called the joint-density-ofstate, which is exactly related to the quantity of available optical transitions. Maybe the authors could refer to this quantity and compute it for the different doping, as done for instance in [Phys. Rev. Lett. 118, 087403 (2017)] or in [Nature Photonics volume 14, pages 183-187 (2020)].

Our reply:
We appreciate for the reviewer's suggestion to improve our explanation on the observed phenomena by using a quantity such as the joint-density-of-states (JDOS  R1a) shows excellent agreement with our experimental data (Fig. 2c). Following the reviewer's comment, we have included the discussion above in the revised manuscript in order to improve our explanation on how forbidden channels contribute to HHG in graphene.

Original comment (4):
3. While the results are interesting, I find the analysis of the results a bit weak, especially for Fig. 2. Following the argument in introduction, one would expect an abrupt change of the yield versus doping at exactly 2µ/Eph=1. However, around this value, almost nothing is happening. I think that the authors should explain this better.

Our reply:
We are grateful for the reviewer' suggestion to improve the analysis of our experimental results on chemical potential dependence of 5 th harmonics intensity. As the reviewer points out, one photon transition does excite tremendous number of charge carriers, which can be blocked when 2|μ|/ = 1. However, only small fraction of these photo-excited carriers participates to generate 5 th harmonics through series of interband or intraband transitions across all electronic states in the momentum space. Instead, carrier excitation via multiphoton transitions can provide more efficient pathways to generate 5 th harmonics as the recent theoretical study [PRB 99, 195407(2019)] demonstrates. In the perturbation regime,  Furthermore, as schematically shown in Fig. R3, our mid-infrared excitation laser has many-cycles of laser-field. Before reaching sufficiently strong laser-field in the middle of the pulse, where supposedly the 5 th harmonic is most strongly generated, first few cycles of laser-field can create large number of background photocarriers which effectively increase doping in graphene even when "static" chemical potential is placed at the Dirac point. For this reason, we believe that we do not observe any abrupt change at 2|μ|/ = 1. We have included the discussion above in the revised manuscript. Estimated temporal profile of electric field for excitation laser pulse. Difference frequency generation serves multi-cycle mid-infrared pulses with pulse width of 120 fs and Eph of 0.28 eV.

Original comment (5):
4. The same figure displays a maximum around 3.4, as nicely shown by the authors for various samples. However, again, this value is not explained. At the moment, we do not know if this value is due to a material's property, a laser property, or a combination of both.
I think that the simulations here could come at help. The authors could also simulate the results of Fig. 2 and try to explain the physical origin of the maximum at 3.4. Which mechanism is dominant for instance. In fact, the same plot as Fig. 2c-d, but decomposed into interband and intraband could really provide some insights here. I would suggest the authors to add this simulations.

Our reply:
We appreciate the reviewer for this important comment which encourages us to provide deeper understanding on 5 th harmonic generation, which is tremendously helpful to improve our manuscript. Following the reviewer's suggestion, we have theoretically calculated from Dirac states in the momentum space which are decomposed to separate contributions from interband and intraband transitions (Fig. R4). Both interband and intraband transitions participate actively in 5 th harmonic generation. Intensity of 5 th harmonic generation is determined by summation of from all the Dirac states. Fig. R4 shows intensity of 5 th harmonic generation as a function of chemical potential (black solid line). The resonance-like profile around at 2|μ|/ = 3.4 is largely originated from interband transitions (red solid line) instead of intraband transition (blue solid line). for the nth order harmonic intensity.
In fact, interband transition via multi-photon excitation are expected to exhibit series of multiple sharp resonance-like profiles at 2|μ|/ = n (n is an integer). In the perturbative regime without considering the scattering process, such profile has been theoretically predicted for by the recent theoretical study [PRB 99, 195407(2019)], which we have drawn in Fig. R2.
However, intense laser excitation creates high density of photocarriers in graphene, which enables rapid electron scattering process. Then, series of sharp resonance-like profiles can be drastically broadened and merged together, forming one resonance-like profile located at 2|μ|/ = non-integer. For example, even in the perturbation regime, the recent experimental and theoretical results [ Fig. 4c in Nature Photonics 12, 430-436 (2018) and New J. Phys. 16, 053014 (2014)] shows that the 3 rd harmonic intensity becomes the highest when 2|μ|/ is away from 2.
In order to investigate such possibility for in our case, we have theoretically calculated 5 th harmonic intensity as a function of chemical potential (Fig.R1) by employing quantum master equation which fully takes into account the scattering process. Fig. R1 shows how massless Dirac states around the Dirac point in the momentum space contributes to generate 5 th harmonic current in graphene under linearly polarized excitation along the x-direction ( ). Dashed lines are constant energy contours on a Dirac cone, which describes electronic states vertically separated by energy (m is an integer). Real and imaginary parts of ( Fig. R1b and R1c, respectively) are most strongly generated from Dirac states in the region between 2 and 4 . Depending on the photon number required for multi-photon transitions and the azimuthal angle, Dirac states generate with characteristic sign and magnitude, which destructively interfere when radiating .
As 2|μ| increases, Pauli blocking sequentially disables contribution from Dirac states starting from near the Dirac point to higher energy. As shown in Fig. R1, increases as the destructive interference is partly eliminated while eventually disappears as all the resonant Dirac states are disabled. Our theoretical calculation (Fig. R1a) also shows that the maximum intensity is exhibited around at 2|μ|/ = 3.4, which is the most common value for 2|μ|/ in our experimental data. We have included the discussion above in the revised manuscript.